Zero Complement is Not Empty
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Theorem
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $S^*$ be the zero complement of $S$.
Then $S^*$ is not empty.
Proof
From Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements, we have:
- $\exists m, n \in S: m \ne n$
That is, there are at least two distinct elements in $S$.
Therefore, there must be at least one element in $S^* = S \setminus \set 0$.
So:
- $S^* = S \setminus \set 0 \ne \O$
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers